// Vec3HemiCos returns a random unit vector chosen on a
// cosine-weighed hemisphere defined by normal
func (r *Random) Vec3HemiCos(normal *maths.Vec3) *maths.Vec3 {
ox := maths.CrossProduct(maths.NewVec3(42, 56, -15), normal)
for math.Abs(ox.Length()) < maths.Epsilon {
ox = maths.CrossProduct(r.Vec3Sphere(), normal)
}
oy := maths.CrossProduct(ox, normal)
ox.Normalise()
oy.Normalise()
u := r.Float01()
radius := math.Sqrt(u)
theta := r.Float02Pi()
vec := normal.Scaled(math.Sqrt(math.Max(0, 1-u)))
vec.Add(ox.Scaled(radius * math.Cos(theta)))
vec.Add(oy.Scaled(radius * math.Sin(theta)))
return vec
}
// Init of Mesh sets the Triangle indices in the Faces array, calculates the bounding box
// and KD tree, sets the surfaceOx and Oy and Cross Products of the sides of each triangle
func (m *Mesh) Init() {
allIndices := make([]int, len(m.Faces))
for i := range allIndices {
allIndices[i] = i
}
m.BoundingBox = m.GetBoundingBox()
m.tree = m.newKDtree(m.BoundingBox, allIndices, 0)
for i := range m.Faces {
triangle := &m.Faces[i]
A := &m.Vertices[triangle.Vertices[0]].Coordinates
B := &m.Vertices[triangle.Vertices[1]].Coordinates
C := &m.Vertices[triangle.Vertices[2]].Coordinates
AB := maths.MinusVectors(B, A)
AC := maths.MinusVectors(C, A)
triangle.AB = AB
triangle.AC = AC
triangle.ABxAC = maths.CrossProduct(AB, AC)
surfaceA := &m.Vertices[triangle.Vertices[0]].UV
surfaceB := &m.Vertices[triangle.Vertices[1]].UV
surfaceC := &m.Vertices[triangle.Vertices[2]].UV
surfaceAB := maths.MinusVectors(surfaceB, surfaceA)
surfaceAC := maths.MinusVectors(surfaceC, surfaceA)
// Solve using Cramer:
// |surfaceAB.X * px + surfaceAX.X *qx + 1 = 0
// |surfaceAB.Y * px + surfaceAX.Y *qx = 0
// and
// surfaceAB.X * py + surfaceAX.X *qy = 0
// surfaceAB.X * py + surfaceAX.X *qy + 1 = 0
px, qx := maths.SolveEquation(surfaceAB, surfaceAC, maths.NewVec3(1, 0, 0))
py, qy := maths.SolveEquation(surfaceAB, surfaceAC, maths.NewVec3(0, 1, 0))
triangle.surfaceOx = maths.AddVectors(AB.Scaled(px), AC.Scaled(qx))
triangle.surfaceOy = maths.AddVectors(AB.Scaled(py), AC.Scaled(qy))
}
}
// IntersectTriangle find whether there's an intersection point between the ray and the triangle
// using barycentric coordinates and calculate the distance
func IntersectTriangle(ray *ray.Ray, A, B, C *maths.Vec3) (bool, float64) {
AB := maths.MinusVectors(B, A)
AC := maths.MinusVectors(C, A)
reverseDirection := ray.Direction.Negative()
distToA := maths.MinusVectors(&ray.Start, A)
ABxAC := maths.CrossProduct(AB, AC)
det := maths.DotProduct(ABxAC, reverseDirection)
reverseDet := 1 / det
if math.Abs(det) < maths.Epsilon {
return false, maths.Inf
}
lambda2 := maths.MixedProduct(distToA, AC, reverseDirection) * reverseDet
lambda3 := maths.MixedProduct(AB, distToA, reverseDirection) * reverseDet
gamma := maths.DotProduct(ABxAC, distToA) * reverseDet
if gamma < 0 {
return false, maths.Inf
}
if lambda2 < 0 || lambda2 > 1 || lambda3 < 0 || lambda3 > 1 || lambda2+lambda3 > 1 {
return false, maths.Inf
}
return true, gamma
}
// IntersectTriangle checks if the bounding box intersects with a triangle
// 1) To have a vertex in the box
// 2) The edge of the triangle intersects with the box
// 3) The middle of the triangle to be inside the box, while the vertices aren't
func (b *BoundingBox) IntersectTriangle(A, B, C *maths.Vec3) bool {
if b.Inside(A) || b.Inside(B) || b.Inside(C) {
return true
}
// we construct the ray from A->B, A->C, etc and check whether it intersects with the box
triangle := [3]*maths.Vec3{A, B, C}
ray := &ray.Ray{}
for rayStart := 0; rayStart < 3; rayStart++ {
for rayEnd := rayStart + 1; rayEnd < 3; rayEnd++ {
ray.Start = *(triangle[rayStart])
ray.Direction = *maths.MinusVectors(triangle[rayEnd], triangle[rayStart])
ray.Init()
// Check if there's intersection between AB and the box (example)
if b.Intersect(ray) {
// we check whether there's intersection between BA and the Box too
// to be sure the edge isn't outside the box
// _____
// | | <----AB there is intersection, but BA----->
// |____|
ray.Start = *triangle[rayEnd]
ray.Direction = *maths.MinusVectors(triangle[rayStart], triangle[rayEnd])
ray.Init()
if b.Intersect(ray) {
return true
}
}
}
}
// we have to check if the inside of the triangle intersects with the box
AB := maths.MinusVectors(B, A)
AC := maths.MinusVectors(C, A)
ABxAC := maths.CrossProduct(AB, AC)
distance := maths.DotProduct(A, ABxAC)
rayEnd := &maths.Vec3{}
for edgeMask := 0; edgeMask < 7; edgeMask++ {
for axis := uint(0); axis < 3; axis++ {
if edgeMask&(1<<axis) != 0 {
continue
}
if edgeMask&1 != 0 {
ray.Start.X = b.MaxVolume[0]
} else {
ray.Start.X = b.MinVolume[0]
}
if edgeMask&2 != 0 {
ray.Start.Y = b.MaxVolume[1]
} else {
ray.Start.Y = b.MinVolume[1]
}
if edgeMask&4 != 0 {
ray.Start.Z = b.MaxVolume[2]
} else {
ray.Start.Z = b.MinVolume[2]
}
*rayEnd = ray.Start
rayEnd.SetDimension(int(axis), b.MaxVolume[axis])
if (maths.DotProduct(&ray.Start, ABxAC)-distance)*(maths.DotProduct(rayEnd, ABxAC)-distance) <= 0 {
ray.Direction = *maths.MinusVectors(rayEnd, &ray.Start)
ray.Init()
intersected, distance := IntersectTriangle(ray, A, B, C)
if intersected && distance < 1.0000001 {
return true
}
}
}
}
return false
}